It is clear that there is little reason to doubt that the New Testament we have was really close or even exactly what what originally written.
There is a way to test the possibility of textual 'perversion' from the original texts. Historians commonly use a method called the bibliographical test. The test is quite simple and is meant to show the relative reliability, as textual preservation is concerned, to other historically accepted manuscripts. To be clear, the test is only meant to show possibility that the text of the original author was changed or not.
First, the you must choose a few manuscripts to compare. We will of course include the new testament gospels (as a whole but we could split it up but that is a lot of work). Then lets throw in Homer's Iliad, from which we gain much knowledge of ancient Greece, and the writtings of Tacitus, from which we gain much knowledge of ancient Rome.
After choosing the manuscripts we compare the earliest known manuscripts and the total number of manuscripts and the similarities between manuscripts. A more complicated version also considers where the manuscript was found, which is significant for ancient times for a number of reasons (but I am disinterested in making this answer too long simply because I will likely not get much rep from it ;)
Bibliographical test of the NT gospels and others
Manuscript Earliest known fragment Earliest known complete copy total number Accuracy to each other
NT Rylands manuscript ~100 years codex Sinaiticus ~400 years over 5800 99.5%
Iliad ~500 years ~500 years over 1200 95%
Tacitus ~800 years None exist (a number of books are missing) less than 10 >90%
On the exact numbers of Tacictus I am being lazy but I am sure that is about right. Check up on it in the various sources I have provided about Tacitus.
The kinds of numbers on Tacitus are typical for ancient manuscripts; The Iliad stands out in a class of its own while the New Testament stand out in a class above that. Having over 5000 manuscripts (mostly Greek) with the earliest completed copy being a mere 400 years in from the events, in addition to high accuracy, is impressive. This is a good indication that there is very little perversion in the texts. To make things better there are 10's of thousands of manuscripts from roughly the 10th century in Latin vulgate that match with the same accuracy, further, translators will tell you that they are pretty true to the Greek manuscripts as well (Ancient Church fathers were very concerned with correct translation). Also, Codex Vaticanus is significant because it is nearly complete and dates 100 years earlier than Sinaiticus. If this is not enough to trust that there is little or no perversion then I don't think anything will be for any ancient text.
There is another important thing to consider, regarding the manuscripts and authorship dates. Most scholars agree that the NT was written in this order. Epistles, synoptic gospels, Acts of the Apostles, Gospel of John, Revelation. It has been highly argued and supporting this claim right now is outside the scope of this question. Consider the following quote from a wikipedia article:
The earliest works which came to be part of the New Testament are the letters of the Apostle Paul.
... the discovery of some New Testament manuscripts and fragments from the 2nd and 3rd centuries, one of which dates as early as 125 [I believe this particular manuscript is from John], disproves a 3rd century date of composition for any book now in the New Testament.
This quote sums up the consensus that is derived from the shear volume of manuscripts, their location found, the significance of codex Sinaiticus and Vaticanus (being complete and nearly complete NT copies), and the widely accepted order of authorship. That consensus is that none of the NT books could have been written later than the 2nd century. Combine this with the bibliographical test and it is a very compelling case to consider the narratives of the Gospel as unchanged from the authors' originals. All that might be left is determining if there was a motive to lie, but that is not your question, but can be answered and has been by many.
* For bibliographical test
* Rylands manuscript
* Codex Sinaiticus
* Codex Vaticanus
* covers Bibliographical test well although it is a biased site
* Annals of Tacitus